Question

A cube made up of wire each of resistance$R$. Then find equivalent resistance across its diagonal

• A. $\frac{5R}{6}$
• B. $\frac{3R}{4}$
• C. $\frac{7R}{12}$
• D. $R$

Answer 1

The correct option is A

$\frac{5R}{6}$

Step 1. Given Data

Resistance of each wire $=R$

Step 2. Finding the Resistance along the diagonal

Along the diagonal$AB$, the circuit can be redrawn as,

Starting from point $A$, we have $3$ different paths along $AC,AG,AF$, these will be in parallel,

then from each of these points, that is $C,E,F$ 7 different paths are possible, making in total $6$ resistances in parallel

Further, the$3$ sides left are $DB,BH,GH$ each meet at the junction of previous parallel paths. going via three separate paths to meet again at $B$

Due to symmetry $C,E,F,$ same potential and $G,D,H,$ same potential, we get,

From the diagram above, We have,

$R_{eq}=\frac{1}{\frac{1}{R}+\frac{1}{R}+\frac{1}{R}}+\frac{1}{\frac{1}{R}+\frac{1}{R}+\frac{1}{R}+\frac{1}{R}+\frac{1}{R}+\frac{1}{R}}+\frac{1}{\frac{1}{R}+\frac{1}{R}+\frac{1}{R}}$

$\Rightarrow R_{eq}=\frac{R}{3}+\frac{R}{6}+\frac{R}{3}$

$\Rightarrow R_{eq}=\frac{5R}{6}$

Hence, the correct option is (A).